Analysis of effective controlled primitive repetitive vectors in Petri net systems

Repetitive vector is one of the important properties of the Petri net basic net, and it can be used to determine the properties of a net language. Unfortunately, a Petri net system usually has infinite effective controlled repetitive vectors. Therefore, it is necessary that all the effective controlled repetitive vectors can be represented with limited entities. Thus, this paper proposes the set of primitive effective controlled repetitive vectors of a Petri net (N,M0), which is denoted with SPECRV(N,M0). After three properties of effective controlled repetitive vectors is analyzed, it is proved that the corresponding properties of all the effective controlled repetitive vectors can be determined, based on some properties of SPECRV(N,M0). The results would play a role in determining the net system dynamic properties such as the liveness and the fairness. Finally, the correctness and the efficiency of the conclusions in the paper are illustrated by a concrete example.